1Cademy - Finding the Domain and Points on the Graph of $$f(x) = \frac{2x-6}{x^2-8x+15}$$ when $$f(x) = 1$$

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Strategy to Solve Equations with Rational Expressions

Example

Finding the Domain and Points on the Graph of $f(x) = \frac{2x-6}{x^2-8x+15}$ when $f(x) = 1$

To analyze the rational function $f(x) = \frac{2x-6}{x^2-8x+15}$ , we can find its domain, solve for $x$ when the function value is $1$ , and identify the corresponding point on its graph.

Step 1: Find the domain. The domain of a rational function consists of all real numbers except those that make the denominator zero. Set the denominator to zero and solve: $x^2 - 8x + 15 = 0$ $(x - 3)(x - 5) = 0$ Using the Zero Product Property: $x - 3 = 0 \quad \text{or} \quad x - 5 = 0$ $x = 3 \quad \text{or} \quad x = 5$ The domain is all real numbers except $x \neq 3$ and $x \neq 5$ .

Step 2: Solve $f(x) = 1$ . Substitute the rational expression for $f(x)$ : $\frac{2x-6}{x^2-8x+15} = 1$ Factor the denominator to find the LCD, which is $(x-3)(x-5)$ : $\frac{2x-6}{(x-3)(x-5)} = 1$ Multiply both sides by the LCD to clear the fraction: $(x-3)(x-5) \cdot \frac{2x-6}{(x-3)(x-5)} = 1 \cdot (x-3)(x-5)$ Simplify the equation: $2x - 6 = x^2 - 8x + 15$ Subtract $2x$ and add $6$ to both sides to set the equation to zero: $0 = x^2 - 10x + 21$ Factor the resulting quadratic equation: $0 = (x - 7)(x - 3)$ Set each factor to zero: $x - 7 = 0 \quad \text{or} \quad x - 3 = 0$ $x = 7 \quad \text{or} \quad x = 3$ Check for extraneous solutions. From Step 1, $x = 3$ is restricted because it makes the denominator zero. Thus, we discard $x = 3$ as an extraneous solution. The only valid solution is $x = 7$ .

Step 3: Find the points on the graph. When $x = 7$ , the function value is $f(x) = 1$ . Therefore, the point $(7, 1)$ lies on the graph of the function.

Updated 2026-04-30

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OpenStax Intermediate Algebra

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